Used to estimate the mean when you have a small sample drawn from a nearly normal population.
Conditions
- Independent observations (\(n < .1N\))
- Nearly normal population distribution (check distribution of sample)
Used to estimate the mean when you have a small sample drawn from a nearly normal population.
The \(t\) has heavier tails than the normal distribution.
The number of parameters that are free to vary, without violating any constraint imposed on it.
\(\mu\)
Since \(\bar{x} = \frac{1}{n}\sum_{i = 1}^n x_i\), one of our observations is contrained, leaving \(n-1\) that are free to vary.
\[ df = n - 1\]
point estimate \(\pm\) margin of error
\[ \bar{x} \pm t^*_{df} \times SE \]
pt(-2.2, df = 18)
## [1] 0.0206
qt(.025, df = 18)
## [1] -2.1
Meet William Sealy Gosset.
Problem: A batch of beer should have a fixed [chemical level related to barley] in order to be of good quality. Can you test a small number of barrels and infer if the entire batch is of good enough quality?